Problem 11 - College Algebra

Question

Express interval in set-builder notation and graph the interval on a number line. \((-\infty, 3)\)

Step-by-Step Solution

Answer

The set-builder notation for the interval \((- \infty, 3)\) is \(x | x < 3\). The graph is a line that starts from 3 (not including 3) and extends to \(- \infty\).

Step 1: Convert to Set-Builder Notation

The set-builder notation for this interval can be expressed as follows: all numbers \(x\), such that \(x\) is less than 3. In terms of mathematical notation, it can be written as \(x | x < 3\).

Step 2: Graph the Interval on a Number Line

On the number line, locate the number 3. Since the interval does not include 3, draw an open circle at the 3. Then, draw a line extending to the left from the open circle to signify all numbers that are less than 3. This line should have an arrow at the end to represent \(- \infty\).

Key Concepts

Interval Notation

Interval notation is a method of denoting the set of all real numbers that fall within a certain range. It is a concise and universally understood system within mathematics. For example, the interval \( (-\infty, 3) \) represents all real numbers less than 3.

To fully understand interval notation, you need to know two basic forms—the open interval and the closed interval. An open interval, such as \( (a, b) \), does not include the endpoints a and b. Meanwhile, a closed interval, such as \( [a, b] \), includes both endpoints. It's important to note that \( -\infty \) and \( +\infty \) are not actual numbers, so intervals including them are always considered open.

When studying interval notation, think of the number line and picture intervals as stretches of road. Just like you wouldn't include the edge of a bridge not meant to be crossed in your path, in an open interval, you don't include the 'edges' of the interval.

Number Line

A number line is a visual representation of numbers on a straight line. Each point on the line corresponds to a number. The number line is essential for visually displaying intervals, understanding the concept of magnitude, and for making inequalities and operations with numbers intuitive. Think of it as a real-life ruler for measuring numerical values.

Graphing Intervals

To graph our previous interval \( (-\infty, 3) \) on a number line, we'd start by drawing a line. We'd then mark a point labeled '3' and draw an open circle—which shows that 3 is not included in the interval. A line extending to the left from the open circle, with an arrow at the end, shows that the interval continues indefinitely in the negative direction. In essence, all points to the left of 3 on the number line are part of the interval.

Inequalities

Inequalities are mathematical expressions indicating that two values are not equal and that one is greater or lesser than the other. They are written using the symbols <, >, ≤, and ≥. For instance, the inequality \(x < 3\) states that \(x\) is any number less than 3.

Understanding Inequalities on the Number Line

Visualizing inequalities on the number line simplifies their meaning. If you imagine standing on the number corresponding to 3 and facing the positive direction, all numbers to your left satisfy the inequality \(x < 3\). If the inequality were \(x \leq 3\), you'd include where you're standing in the set, which is usually marked with a closed dot on the number line to show inclusion.